Line-closed matroids, quadratic algebras, and formal arrangements
Michael Falk
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21 pages, 6 figures. To appear in Advances in Applied Mathematics. Text has been shortened and substantially revised to clarify status of problem and several other important points. Theorem 2.4 has been made more precise and Corollary 2.20 has been strengthened
Let $G$ be a matroid on ground set \A. The Orlik-Solomon algebra $A(G)$ is the quotient of the exterior algebra \E on \A by the ideal \I generated by circuit boundaries. The quadratic closure $\bar{A}(G)$ of $A(G)$ is the quotient of \E by the ideal generated by the degree-two component of \I. We introduce the notion of \nbb set in $G$, determined by a linear order on \A, and show that the corresponding monomials are linearly independent in the quadratic closure $\bar{A}(G)$. As a consequence, $A(G)$ is a quadratic algebra only if $G$ is line-closed. An example of S.~Yuzvinsky proves the converse false. These results generalize to the degree $r$ closure of $\A(G)$.
The motivation for studying line-closed matroids grew out of the study of formal arrangements. This is a geometric condition necessary for \A to be free and for the complement $M$ of \A to be a $K(π,1)$ space. Formality of \A is also necessary for $A(G)$ to be a quadratic algebra. We clarify the relationship between formality, line-closure, and other matroidal conditions related to formality. We give examples to show that line-closure of $G$ is not necessary or sufficient for $M$ to be a $K(π,1)$, or for \A to be free.
We study the combinatorial properties of a tropical hyperplane arrangement. We define tropical oriented matroids, and prove that they share many of the properties of ordinary oriented matroids. We show that a tropical oriented matroid determines a subdivision of a product of two simplices, and conjecture that this correspondence is a bijection.
For a natural number $c$, a $c$-arrangement is an arrangement of dimension $c$ subspaces satisfying the following condition: the sum of any subset of the subspaces has dimension a multiple of $c$. Matroids arising as normalized rank functions of $c$-arrangements are also known as multilinear matroids. We prove that it is algorithmically undecidable whether there exists a $c$ such that a given matroid has a $c$-arrangement representation, or equivalently whether the matroid is multilinear. It follows that certain network coding problems are also undecidable. In the proof, we introduce a general…
It is known that there exist hyperplane arrangements with same underlying matroid that admit non-homotopy equivalent complement manifolds. In this work we show that, in any rank, complex central hyperplane arrangements with up to 7 hyperplanes and same underlying matroid are isotopic. In particular, the diffeomorphism type of the complement manifold and the Milnor fiber and fibration of these arrangements are combinatorially determined, i.e., they depend uniquely on the underlying matroid. To do this, we associate to every such matroid a topological space, that we call the reduced realization …
A matroid is a machine capturing linearity of mathematical objects and producing combinatorial structures. Matroid structure arises everywhere since linearity is a ubiquitous concept. One natural way to obtain matroids is by considering hyperplane arrangements, which give rise to convex polytopes called matroid polytopes. Much research has been conducted on these three areas: matroids, matroid polytopes, and hyperplane arrangements. However, substantial gaps in our knowledge remain, and the correspondence diagram between those areas needs to be more extensive. For instance, currently, there is…
To every realizable oriented matroid there corresponds an arrangement of real hyperplanes. The homeomorphism type of the complexified complement of such an arrangement is completely determined by the oriented matroid. In this paper we study arrangements of pseudohyperplanes; they correspond to non-realizable oriented matroids. These arrangements arise as a consequence of the Folkman-Lawrence topological representation theorem. We propose a generalization of the complexification process in this context. In particular we construct a space naturally associated with these pseudo-arrangements which…
Swartz proved that any matroid can be realized as the intersection lattice of an arrangement of codimension one homotopy spheres on a sphere. This was an unexpected extension from the oriented matroid case, but unfortunately the construction is not explicit. Anderson later provided an explicit construction, but had to use cell complexes of high dimensions that are homotopy equivalent to lower dimensional spheres. Using diagrams of spaces we give an explicit construction of arrangements in the right dimensions. Swartz asked if it is possible to arrange spheres of codimension two, and we provide…
We introduce the notion of an arithmetic matroid, whose main example is given by a list of elements of a finitely generated abelian group. In particular we study the representability of its dual, providing an extension of the Gale duality to this setting. Guided by the geometry of generalized toric arrangements, we provide a combinatorial interpretation of the associated arithmetic Tutte polynomial, which can be seen as a generalization of Crapo's formula for the classical Tutte polynomial.
A phased matroid is a matroid with additional structure which plays the same role for complex vector arrangements that oriented matroids play for real vector arrangements. The realization space of an oriented (resp., phased) matroid is the space of vector arrangements in $\mathbb R^n$ (resp., $\mathbb C^n$) that correspond to oriented (resp., phased) matroid, modulo a change of coordinates. According to Mnëv's Universality Theorem, the realization spaces of uniform oriented matroids with rank greater than or equal to $3$ can be as complicated as any open semi-algebraic variety. In contrast, un…
We verify the Rota-Heron-Welsh conjecture for matroids realizable as c-arrangements: the coefficients of the characteristic polynomial of the associated matroid are log-concave. This family of matroids strictly contains that of complex hyperplane arrangements. Our proof combines the study of intrinsic volumes of certain extensions of arrangements and the Levy--Milman measure concentration phenomenon on realization spaces of arrangements.
We construct minimal cellular resolutions of squarefree monomial ideals arising from hyperplane arrangements, matroids and oriented matroids. These are Stanley-Reisner ideals of complexes of independent sets, and of triangulations of Lawrence matroid polytopes. Our resolution provides a cellular realization of Stanley's formula for their Betti numbers. For unimodular matroids our resolutions are related to hyperplane arrangements on tori, and we recover the resolutions constructed by Bayer, Popescu and Sturmfels. We resolve the combinatorial problems posed in their paper by computing Mobius in…
A catalogue of simplicial hyperplane arrangements was first given by Grünbaum in 1971. These arrangements naturally generalize finite Coxeter arrangements and the weak order through the poset of regions. For simplicial arrangements, posets of regions are in fact lattices. We update Grünbaum's catalogue, providing normals and invariants for all known sporadic simplicial arrangements with up to 37 lines. The weak order is known to be congruence normal, and congruence normality for simplicial arrangements can be determined using polyhedral cones called shards. In this article, we provide addition…
Given an $\Bbb{F}$-represented matroid $(M,ρ)$ with the ground set $[m]$, the representation $ρ$ naturally defines a hyperplane arrangement $\mathcal{A}_ρ$. We will study its parallel translates $\mathcal{A}_{ρ,{g}}$ of $\mathcal{A}_ρ$ for all ${ g}\in \mathbb{F}^m$. Its intersection semi-lattices $L(\mathcal{A}_{ρ,{ g}})$ and the characteristic polynomials $χ(\mathcal{A}_{ρ,{ g}},t)$ will be classified by the intersection lattice of the derived arrangement $\mathcal{A}_{δρ}$, which is a hyperplane arrangement associated with the derived matroid $(δM,δρ)$ and also known as the discriminantal a…
There is a one-to-one correspondence between geometric lattices and the intersection lattices of arrangements of homotopy spheres. When the arrangements are essential and fully partitioned, Zaslavsky's enumeration of the cells of the arrangement still holds. An application of the theory shows that all minimal cellular resolutions of matroid Steiner ideals are bounded subcomplexes of homotopy sphere arrangements of the given matroid. As a result the Betti numbers of the ideal are computed and seen to be equivalent to Stanley's formula in the special case of face ideals of independence complexes…
The construction of the Varchenko matrix for hyperplane arrangements, first introduced by Alexandre Varchenko, extends naturally to oriented matroids. In this paper, we generalize the theorem of Gao and Zhang by proving that the Varchenko matrix of an oriented matroid has a diagonal form if and only if the pseudohyperplane arrangement corresponding to the oriented matroid is in semigeneral position, i.e. it does not contain a degeneracy. Furthermore, we show that the Varchenko matrix of a pseudoline arrangement has a block diagonal form. This also provides an alternative combinatorial proof fo…
A classic problem in matroid theory is to find subspace arrangements, specifically hyperplane and pseudosphere arrangements, whose intersection posets are isomorphic to a prescribed geometric lattice. Engström recently showed how to construct an infinite family of such subspace arrangements, indexed by the set of finite regular CW complexes. In this note, we compute the face numbers of these representations (in terms of the face numbers of the indexing complexes) and give upper bounds on the total number of faces in these objects. In particular, we show that, for a fixed rank, the total number…
Ardila and Develin's paper on tropical oriented hyperplane arrangements and tropical oriented matroids defines tropical oriented matroids and conjectures a bijection between them and triangulations of products of simplices $Δ_{n-1} \times Δ_{d-1}$. Oh and Yoo recently confirmed this conjecture; however, neither group addressed the case of hyperplanes that are not in generic position. These non-generic arrangements do not correspond to tropical oriented matroids, but they encode information about subdivisions of $Δ_{n-1} \times Δ_{d-1}$. This note considers the non-generic case and presents som…
The $OS$ algebra $A$ of a matroid $M$ is a graded algebra related to the Whitney homology of the lattice of flats of $M$. In case $M$ is the underlying matroid of a hyperplane arrangement \A in $\C^r$, $A$ is isomorphic to the cohomology algebra of the complement $\C^r\setminus \bigcup \A.$ Few examples are known of pairs of arrangements with non-isomorphic matroids but isomorphic $OS$ algebras. In all known examples, the Tutte polynomials are identical, and the complements are homotopy equivalent but not homeomorphic. We construct, for any given simple matroid $M_0$, a pair of infinite famili…
Tropical varieties play an important role in algebraic geometry. The Bergman complex B(M) and the positive Bergman complex B+(M) of an oriented matroid M generalize to matroids the notions of the tropical variety and positive tropical variety associated to a linear ideal. Our main result is that if A is a Coxeter arrangement of type Phi with corresponding oriented matroid M_Phi, then B+(M_Phi) is dual to the graph associahedron of type Phi, and B(M_Phi) equals the nested set complex of A. In addition, we prove that for any orientable matroid M, one can find |mu(M)| different reorientations of …
In 1993, Csima and Sawyer proved that in a non-pencil arrangement of n pseudolines, there are at least $\frac{6}{13}n$ simple points of intersection. Since pseudoline arrangements are the topological representations of reorientation classes of oriented matroids of rank $3$, in this paper, we will use this result to prove by induction that an oriented paving matroid of rank $r \ge 3$ on $n$ elements, where $n \geq 5+ r$, has at least $\frac{12}{13(r-1)} \binom{n}{r-2}$ independent hyperplanes, yielding a new necessary condition for a paving matroid to be orientable.
We define a partial ordering on the set Q = Q(M) of pairs of topes of an oriented matroid M, and show the geometric realization |Q| of the order complex of Q has the same homotopy type as the Salvetti complex of M. For any element e of the ground set, the complex |Qe| associated to the rank-one oriented matroid on {e} has the homotopy type of the circle. There is a natural free simplicial action of Z4 on |Q|, with orbit space isomorphic to the order complex of the poset Q(M,e) associated to the pointed (or affine) oriented matroid (M,e). If M is the oriented matroid of an arrangement A of line…
In this article, we study freeness of hyperplane arrangements. One of the most investigated arrangement is a graphic arrangement. Stanley proved that a graphic arrangement is free if and only if the corresponding graph is chordal and Dirac showed that a graph is chordal if and only if the graph is obtained by "gluing" complete graphs. We will generalize Dirac's construction to simple matroids with modular joins introduced by Ziegler and show that every arrangement whose associated matroid is constructed in the manner mentioned above is divisionally free. Moreover, we apply the result to arrang…
Zaslavsky conjectures that the bounded complex of a simple hyperplane arrangement is homeomorphic to a ball. We prove this conjecture for the more general uniform affine oriented matroids.
For all positive integers $t$ exceeding one, a matroid has the cyclic $(t-1,t)$-property if its ground set has a cyclic ordering $σ$ such that every set of $t-1$ consecutive elements in $σ$ is contained in a $t$-element circuit and $t$-element cocircuit. We show that if $M$ has the cyclic $(t-1,t)$-property and $|E(M)|$ is sufficiently large, then these $t$-element circuits and $t$-element cocircuits are arranged in a prescribed way in $σ$, which, for odd $t$, is analogous to how 3-element circuits and cocircuits appear in wheels and whirls, and, for even $t$, is analogous to how 4-element cir…
The Tutte polynomial is a fundamental invariant associated to a graph, matroid, vector arrangement, or hyperplane arrangement. This short survey focuses on some of the most important results on Tutte polynomials of hyperplane arrangements. We show that many enumerative, algebraic, geometric, and topological invariants of a hyperplane arrangement can be expressed in terms of its Tutte polynomial. We also show that, even if one is only interested in computing the Tutte polynomial of a graph or a matroid, the theory of hyperplane arrangements provides a powerful finite field method for this compu…
We use techniques from Gromov-Witten theory to construct new invariants of matroids taking value in the Chow groups of spaces of rational curves in the permutohedral toric variety. When the matroid is realizable by a complex hyperplane arrangement, our invariants coincide with virtual fundamental classes used to define the logarithmic Gromov-Witten theory of wonderful models of arrangement complements, for any logarithmic structure supported on the wonderful boundary. When the boundary is empty, this implies that the quantum cohomology ring of a hyperplane arrangement's wonderful model is a co…
An arrangement of hyperplanes is called formal, if the relations between the hyperplanes are generated by relations in codimension 2. Formality is not a combinatorial property, raising the question for a characterization for combinatorial formality. A sufficient condition for this is if the underlying matroid has no proper lift with the same points and lines. We present an example of a matroid with such a lift but no non-formal realization, thus showing that above condition is not necessary for combinatorial formality.
Varchenko introduced in 1993 a distance function on the chambers of a hyperplane arrangement that gave rise to a determinant whose entry in position $(C, D)$ is the distance between the chambers $C$ and $D$, and computed that determinant. In 2017, Aguiar and Mahajan provided a generalization of that distance function, and computed the corresponding determinant. This article extends their distance function to the topes of an oriented matroid, and computes the determinant thus defined. Oriented matroids have the nice property to be abstractions of some mathematical structures including hyperplan…
Let A be a (central) arrangement of hyperplanes in a finite dimension complex vector space V. Let M(A) be the dependence matroid determined by A. The Orlik-Solomon algebra OS(M) of a matroid M is the exterior algebra on the points modulo the ideal generated by circuit boundaries. The algebra OS(M) is isomorphic to the cohomology algebra of the complement in V of the union of the hyperplanes of A. The Tutte polynomial T(x,y) of M is a powerful invariant of the matroid M. When M(A) is a rank three matroid and A is the complexification of a real arrangement, we prove that OS(M) determines T(x,y).…
We show that the integral cohomology algebra of the complement of a toric arrangement is not determined by the poset of layers. Moreover, the rational cohomology algebra is not determined by the arithmetic matroid (however it is determined by the poset of layers).
In this paper we describe a parallel algorithm for generating all non-isomorphic rank $3$ simple matroids with a given multiplicity vector. We apply our implementation in the HPC version of GAP to generate all rank $3$ simple matroids with at most $14$ atoms and a splitting characteristic polynomial. We have stored the resulting matroids alongside with various useful invariants in a publicly available, ArangoDB-powered database. As a byproduct we show that the smallest divisionally free rank $3$ arrangement which is not inductively free has $14$ hyperplanes and exists in all characteristics di…
We investigate the line arrangement that results from intersecting d complete flags in C^n. We give a combinatorial description of the matroid T_{n,d} that keeps track of the linear dependence relations among these lines. We prove that the bases of the matroid T_{n,3} characterize the triangles with holes which can be tiled with unit rhombi. More generally, we provide evidence for a conjectural connection between the matroid T_{n,d}, the triangulations of the product of simplices Delta_{n-1} x Δ_{d-1}, and the arrangements of d tropical hyperplanes in tropical (n-1)-space. Our work provides a …
The interior and exterior activities of bases of a matroid are well-known notions that for instance permit one to define the Tutte polynomial. Recently, we have discovered correspondences between the regions of gainic hyperplane arrangements and coloredlabeled rooted trees. Here we define a general activity theory that applies in particular to no-broken circuit (NBC) sets and labeled colored trees. The special case of activity \textsf{0} was our motivating case. As a consequence, in a gainic hyperplane arrangement the number of bounded regions is equal to the number of the corresponding colore…
We study arrangements of slightly skewed tropical hyperplanes, called blades by A. Ocneanu, on the vertices of a hypersimplex $Δ_{k,n}$, and we investigate the resulting induced polytopal subdivisions. We show that placing a blade on a vertex $e_J$ induces an $\ell$-split matroid subdivision of $Δ_{k,n}$, where $\ell$ is the number of cyclic intervals in the $k$-element subset $J$. We prove that a given collection of $k$-element subsets is weakly separated, in the sense of the work of Leclerc and Zelevinsky on quasicommuting families of quantum minors, if and only if the arrangement of the bla…
We study Stanley-Reisner ideals of broken circuits complexes and characterize those ones admitting a linear resolution or being complete intersections. These results will then be used to characterize arrangements whose Orlik-Terao ideal has the same properties. As an application, we improve a result of Wilf on upper bounds for the coefficients of the chromatic polynomial of a maximal planar graph. We also show that for an ordered matroid with disjoint minimal broken circuits, the supersolvability of the matroid is equivalent to the Koszulness of its Orlik-Solomon algebra.
Let k be a field of characteristic zero. We consider graded subalgebras A of k[x_1,...,x_m]/(x_1^2,...,x_m^2) generated by d linearly independant linear forms. Representations of matroids over k provide a natural description of the structure of these algebras. In return, the numerical properties of the Hilbert function of A yield some information about the Tutte polynomial of the corresponding matroid. Isomorphism classes of these algebras correspond to equivalence classes of hyperplane arrangements under the action of the general linear group.
In this paper we build an Orlik-Solomon model for the canonical gradation of the cohomology algebra with integer coefficients of the complement of a toric arrangement. We give some results on the uniqueness of the representation of arithmetic matroids, in order to discuss how the Orlik-Solomon model depends on the poset of layers. The analysis of discriminantal toric arrangements permits us to isolate certain conditions under which two toric arrangements have diffeomorphic complements. We also give combinatorial conditions determining whether the cohomology algebra is generated in degree one.
We compute the cohomology ring of the complement of a toric arrangement with integer coefficients and investigate its dependency from the arrangement's combinatorial data. To this end, we study a morphism of spectral sequences associated to certain combinatorially defined subcomplexes of the toric Salvetti category in the complexified case, and use a technical argument in order to extend the results to full generality. As a byproduct we obtain: -a "combinatorial" version of Brieskorn's lemma in terms of Salvetti complexes of complexified arrangements, -a uniqueness result for realizations of a…
We expand upon the notion of equivariant log concavity, and make equivariant log concavity conjectures for Orlik--Solomon algebras of matroids, Cordovil algebras of oriented matroids, and Orlik--Terao algebras of hyperplane arrangements. In the case of the Coxeter arrangement for the Lie algebra $\mathfrak{sl}_n$, we exploit the theory of representation stability to give computer assisted proofs of these conjectures in low degree.
The active bijection for oriented matroids (and real hyperplane arrangements, and graphs, as particular cases) is introduced and investigated by the authors in a series of papers. Given any oriented matroid defined on a linearly ordered ground set, we exhibit one particularité of its bases, which we call its active basis, with remarkable properties. It preserves activities (for oriented matroids in the sense of Las Vergnas, for matroid bases in the sense of Tutte), as well as some active partitions of the ground set associated with oriented matroids and matroid bases. It yields a canonical bij…
We introduce a greedy algorithm optimizing arrangements of lines with respect to a property. We apply this algorithm to the case of simpliciality: it recovers all known simplicial arrangements of lines in a very short time and also produces a yet unknown simplicial arrangement with 35 lines. We compute a (certainly incomplete) database of combinatorially simplicial complex arrangements of hyperplanes with up to 50 lines. Surprisingly, it contains several examples whose matroids have an infinite space of realizations up to projectivities.
